Zerlegung von Quadraten und ???? Zerlegung von Quadraten und ????

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Zerlegung von Quadraten und ????

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1

1 x

x

y

z z y

2x+y=1 y+z=x 3z=y

15/11

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11

11 6

1 4

 11

4

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1 1

11 3

4 2 4

  



11

11 4

11

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11

11 4

11

3

1 1 1 1

1 11 3 1 1

2 2 2

4 1

4 4 1

3 3

   

  



4

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1 1 1 1

11 3 1 1

2 2 2

4 1

4 4 1

3 3

      

  



11

11 4

3

1 4

4

1 1

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15/9

9

9 6

3

Kettenbruchentwicklung und ggT:

Die Länge des kleinsten Quadrats ist der ggT

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r_n hat fraktionalen

Anteil

a_n:= ganzzahliger Anteil von r_n Ja

Nein

z:=1/(r_n– a_n)

a_n:= r_n Ende

z := x, _,n:=0

Euklidischer Algorithmus

r_n:= z

n:=n+1 Gegeben x

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Kettenbrüche und ähnliche Rechtecke

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Kettenbrüche und ähnliche Rechtecke

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x

x y

y

x-y

[0;1,1,1,...]

x y y

x x y y

   



[1;1,1,...]

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x

x y

y

x-2y

[0; 2,1, 2,1,...]

2

x y y

x x y y

   



[1; 2,1, 2,1, 2,...]

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x

x y

y

x-3y

[0;3,1,3,1,...]

3

x y y

x x y y

   



[1;3,1,3,1,3,...]

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x

x y

y

x-ny

[0; ,1, ,1,...]

x y y

y n n

x x ny

   



[1; ,1, ,1, ,...] n n n

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Was sind die Gleichungen für:

[1;1,2,1,1,2,1,1,2,…]

[1;1,3,1,1,3,1,1,3,…]

[1;2,3,1,2,3,1,2,3,…]

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x

x y

y x

x-2y

2 [0; 2, 2, 2,...]

2

x y y

x x y y

   



[2; 2, 2, 2,...]

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x

x y

y

x-3y

3 [0;3,3,3,...]

3

x y y

x x y y

   



[3;3,3,3,...]

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nx y y x x ny

 



2

1 0

y  ny  

( ) 1

y n y  

1 [0; , , ,...]

y y n n n

 n y  



[ ; , , ,...] n n n n

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x rational:

x kann in der Form m/n geschrieben werden; m und n natürliche Zahlen x hat schließlich-periodische Entwicklung bezüglich jeder Basis

x hat abbrechende Kettenbruchentwicklung x irrational:

x nicht als Quotient zweier natürlicher Zahlen als m/n schreibbar x keine Periodizität in der Entwicklung bezüglich jeder Basis

x hat Kettenbruchentwicklung, die nicht abbricht

Wenn x algebraisch von der Ordnung 2 (und irrational), dann hat x eine schließlich-periodische Kettenbruchentwicklung.

Es gilt auch die Umkehrung!

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n [ a; Period ]

√² 1; 2

√³ 1; 1,2

√⁴ 2;

√⁵ 2; 4

√⁶ 2; 2,4

√⁷ 2; 1,1,1,4

√⁸ 2; 1,4

√⁹ 3;

√¹

0 3; 6

√¹

1 3; 3,6

√¹

2 3; 2,6

√¹

3 3; 1,1,1,1,6

√¹

4 3; 1,2,1,6

√¹

5 3; 1,6

√¹

6 4;

√¹

7 4; 8

√¹

8 4; 4,8

√¹

9 4; 2,1,3,1,2,8

√²

0 4; 2,8

√²

1 4; 1,1,2,1,1,8

√²

2 4; 1,2,4,2,1,8

√²

3 4; 1,3,1,8

√²

4 4; 1,8

√²

5 5;

√²

6 5; 10

√²

7 5; 5,10

√²

8 5; 3,2,3,10

√²

9 5; 2,1,1,2,10

√³

0 5; 2,10

√³

1 5; 1,1,3,5,3,1,1,10

√³

2 5; 1,1,1,10

√³

3 5; 1,2,1,10

√³

4 5; 1,4,1,10

√³

5 5; 1,10

√³

6 6;

√³

7 6; 12

√³

8 6; 6,12

√³

9 6; 4,12

√⁴

0 6; 3,12

√⁴

1 6; 2,2,12

√⁴

2 6; 2,12

√⁴

3 6; 1,1,3,1,5,1,3,1,1,12

√⁴

4 6; 1,1,1,2,1,1,1,12

√⁴

5 6; 1,2,2,2,1,12

√⁴

6

6;

1,3,1,1,2,6,2,1,1,3,1,1 2

√⁴

7 6; 1,5,1,12

√⁴

8 6; 1,12

√⁴

9 7;

√⁵

0 7; 14

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n [ a; Period ]

√⁵

1 7; 7,14

√⁵

2 7; 4,1,2,1,4,14

√⁵

3 7; 3,1,1,3,14

√⁵

4 7; 2,1,6,1,2,14

√⁵

5 7; 2,2,2,14

√⁵

6 7; 2,14

√⁵

7 7; 1,1,4,1,1,14

√⁵

8 7; 1,1,1,1,1,1,14

√⁵

9 7; 1,2,7,2,1,14

√⁶

0 7; 1,2,1,14

√⁶

1 7; 1,4,3,1,2,2,1,3,4,1,14

√⁶

2 7; 1,6,1,14

√⁶

3 7; 1,14

√⁶

4 8;

√⁶

5 8; 16

√⁶

6 8; 8,16

√⁶

7 8; 5,2,1,1,7,1,1,2,5,16

√⁶

8 8; 4,16

√⁶

9 8; 3,3,1,4,1,3,3,16

√⁷

0 8; 2,1,2,1,2,16

√⁷

1 8; 2,2,1,7,1,2,2,16

√⁷

2 8; 2,16

√⁷

3 8; 1,1,5,5,1,1,16

√⁷

4 8; 1,1,1,1,16

√⁷

5 8; 1,1,1,16

√⁷

6 8; 1,2,1,1,5,4,5,1,1,2,1,16

√⁷

7 8; 1,3,2,3,1,16

√⁷

8 8; 1,4,1,16

√⁷

9 8; 1,7,1,16

√⁸

0 8; 1,16

√⁸

1 9;

√⁸

2 9; 18

√⁸

3 9; 9,18

√⁸

4 9; 6,18

√⁸

5 9; 4,1,1,4,18

√⁸

6 9; 3,1,1,1,8,1,1,1,3,18

√⁸

7 9; 3,18

√⁸

8 9; 2,1,1,1,2,18

√⁸

9 9; 2,3,3,2,18

√⁹

0 9; 2,18

√⁹

1 9; 1,1,5,1,5,1,1,18

√⁹

2 9; 1,1,2,4,2,1,1,18

√⁹

3 9; 1,1,1,4,6,4,1,1,1,18

√⁹

4

9;

1,2,3,1,1,5,1,8,1,5,1,1,3,2,1, 18

√⁹

5 9; 1,2,1,18

√⁹

6 9; 1,3,1,18

√⁹

7 9; 1,5,1,1,1,1,1,1,5,1,18

√⁹

8 9; 1,8,1,18

√⁹

9 9; 1,18

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Realisierung des Euklidischen Algorithmus mit Microsoft Excel

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Kettenbruchentwicklungen von

, , , tanh(1), tan(1),...

2

e e 

http://en.wikipedia.org/wiki/Continued_fraction

http://wims.unice.fr/wims/wims.cgi?lang=en&module=tool%2Fnumber%2Fcontfrac.en&cmd=new

http://mathworld.wolfram.com/ContinuedFraction.html

Calculator:

Gute Seiten:

http://home.att.net/~numericana/answer/fractions.htm#continued

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html#intro

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Contfrac

--- Help [Back] ---

Examples of expressions, and how to enter them.

For the expression: You may type: Which gives:

^{pi^2-3*e} 1.7147589...

sqrt(2)+5^(1/3) 3.1241895...

4⁶-3⁶-2⁶ 4^6-3^6-2^6 3303

2²²-10! ^{2^22-10!} 565504

(3⁵-1)(2⁵-1)-1 (3^5-1)*(2^5-1)-1 7501 (15+77-2)/(2^3*3^5-1) 90/1943 More advanced examples .

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Contfrac

--- Help [Back] ---

Examples of expressions, and how to enter them.

For the expression: You may type: Which gives:

ppcm(15,70)-pgcd(21,33) lcm(15,70)-gcd(21,33) 207 sum(n=1,10,n^2+n) 440 prod(n=1,10,n^2/

(n^2+1))

binomial(30,12) 86493225 integral part of e⁴ truncate(exp(4)) 54

2^(2^(2^2))-8! 25216

root of x²+x-1

between 0 and 1 (golden ratio) solve(x=0,1,x^2+x-1) 0.618033988...

Elementary examples .

For more information on the functions and their names, please consult the manual of pari.

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 = ½ (1+5) [1; 1, 1, 1, 1, 1, 1, 1, 1, ...

½ [k+(k²+4)] [k; k, k, k, k, k, k, k, k, ...

2 [1; 2, 2, 2, 2, 2, 2, 2, 2, ...

3 [1; 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...

5 [2; 4, 4, 4, 4, 4, 4, 4, 4, ...

7 [2; 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, ...

41 [6; 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, ...

e = exp(1) [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, ... 2n+2, 1, 1, ...

e = exp(1/2) [1; 1, 1, 1, 5, 1, 1, 9, 1, 1, 13, 1, 1, 17, 1, 1, ... 4n+1, 1, 1, ...

exp(1/3) [1; 2, 1, 1, 8, 1, 1, 14, 1, 1, 20, 1, 1, 26, 1, 1, ... 6n+2, 1, 1 ...

exp(1/k) [1; k-1, 1, 1, 3k-1, 1, 1, 5k-1, 1, 1, 7k-1, ... (2n+1)k-1, 1, 1 ...

e² = exp(2) [7; 2, 1, 1, 3, 18, 5, 1, 1, 6, ... 12n+6, 3n+2, 1, 1, 3n+3 ...

exp(2/3) [1; 1, 18, 7, 1, 1, 10, ... 36n+18, 9n+7, 1, 1, 9n+10 ...

exp(2/5) [1; 2, 30, 12, 1, 1, 17, ... 60n+30, 15n+12, 1, 1, 15n+17 ...

exp(2/7) [1; 3, 42, 17, 1, 1, 24, ... 84n+42, 21n+17, 1, 1, 21n+24 ...

exp(2/(2k+1)) [1; k, ... (24k+12)n+12k+6, (6k+3)n+5k+2, 1, 1, (6k+3)n+7k+3

...

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tanh(1) =

(e²-1)/(e²+1) [0; 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, ... (2n+1) ...

tanh(1/k) [0; k, 3k, 5k, 7k, 9k, 11k, 13k, 15k, 17k, 19k, ... (2n+1)k ...

tan(1) [1; 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13, 1, 15, 1, ... 2n+1, 1, ...

tan(1/2) [0; 1, 1, 4, 1, 8, 1, 12, 1, 16, 1, 20, 1, 24, 1, 28, 1, ... 4n, 1, ...

tan(1/k) [0; k-1, 1, 3k-2, 1, 5k-2, 1, 7k-2, 1, 9k-2, 1, ... (2n+1)k-2,1, ...